On a problem of Nagata related to Zariski's problem
type: Item
instance of: scholarly article
title: On a problem of Nagata related to Zariski's problem
zbMATH Open document ID: 0944.13013
author: Tetsushi Ogoma
published in: Osaka Journal of Mathematics
publication date: 1999-01-24T00:00:00Z
review text: \textit{M. Nagata} [in Osaka J. Math. 33, No. 4, 997-1002 (1996; see the preceding review Zbl 0944.13012)] has introduced the notion of divisorial valuation ring. Namely, let \(B\) be a noetherian (integral) domain. We say that \(D\) is a divisorial valuation ring over \(B\) if \(D\) is some localization of the derived normal ring (i.e. the integral closure in its field of quotients) of an affine ring over \(B\) by a height one prime ideal. Related to his work on Zariski's problem (see op.cit.), he proved that If \(B\) is a noetherian integral domain for which the following property holds: (*) Let \(K\) be the quotient field of \(B\). Then for every divisorial valuation ring \(D\) over \(B\), the intersection of \(D\) with \(K\) is a divisorial valuation ring over \(B\), or is equal to \(K\). Then a ring \(R\) over \(B\) is the intersection of the integral closure of an affine ring over \(B\) with a function field over \(B\), if and only if \(R\) is the \(I\)-transform of the integral closure of some affine ring over \(B\). Motivated by this result he asked the question: What is the class of noetherian integral domains for which the condition (*) above holds ? In the paper under review, the author shows (theorem 2.3), using localization and properties of unibranched local domains, that the property (*) holds for every noetherian domain, giving a simpler restatement of Nagata's results.
reviewed by: Philippe Elbaz-Vincent
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13B30
zbMATH DE Number: 1217815
zbMATH Keywords: localization
zbMATH Keywords: divisorial valuation ring
zbMATH Keywords: Nagata's problem
zbMATH Keywords: Zariski's problem
zbMATH Keywords: unibranched local domain
zbMATH Keywords: ideal transform
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1217815